Problem: Factor the following expression: $-6$ $x^2+$ $17$ $x$ $-10$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-6)}{(-10)} &=& 60 \\ {a} + {b} &=& & & {17} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $60$ and add them together. The factors that add up to ${17}$ will be your ${a}$ and ${b}$ When ${a}$ is ${5}$ and ${b}$ is ${12}$ $ \begin{eqnarray} {ab} &=& ({5})({12}) &=& 60 \\ {a} + {b} &=& {5} + {12} &=& 17 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-6}x^2 +{5}x +{12}x {-10} $ Group the terms so that there is a common factor in each group: $ ({-6}x^2 +{5}x) + ({12}x {-10}) $ Factor out the common factors: $ x(-6x + 5) - 2(-6x + 5) $ Notice how $(-6x + 5)$ has become a common factor. Factor this out to find the answer. $(-6x + 5)(x - 2)$